\section{1.9} 
\begin{frame}[allowframebreaks]{1.9. }

\vspace{-0.4cm}

1.9. Local good filtrations, Characteristic variety. 

On $\mathcal{D}_X$, we have the usual filtration by the degree, defined by induction on $m \in \mathbb{N}$ by:
\begin{equation}
\mathcal{D}_X(0) = \mathcal{O}_X
\end{equation}
\begin{equation}
\mathcal{D}_X(m) = \{ P \in \mathcal{D}_X \mid [P, \mathcal{O}_X] \subset \mathcal{D}_X(m-1) \} \quad (m \ge 1)
\end{equation}

More explicitly, in an affine neighborhood $Y$ with local coordinates $(x_i, \partial_i)$, $\mathcal{D}(Y)(m) = \mathcal{D}_X(m)(Y)$ consists of the $P$'s of the form
\begin{equation}
P = \sum_{|\alpha| \le m} f_\alpha \cdot \partial^\alpha, \quad \text{with } f_\alpha \in \mathcal{O}(Y).
\end{equation}

As in the case of the Weyl algebra, we see that
\begin{equation}
\mathrm{Gr}\,\mathcal{D}(Y) = \mathcal{O}(Y)[\xi_1, \ldots, \xi_n] \quad (n = \dim Y),
\end{equation}
where the $\xi_i$ are commuting variables representing the $\partial_i$'s. 

This is the direct image of the coordinate ring of the cotangent bundle $T^*(Y)$ of $Y$ under the natural projection. 

Therefore
\begin{equation}
\mathrm{Gr}\,\mathcal{D}_X = \tau_* (\mathcal{O}_{T^*(X)}),
\end{equation}
where $\tau: T^*(X) \to X$ is the natural projection.

The ring $\mathcal{D}(Y)$ satisfies the assumptions (i), (ii) imposed on $R$ in (V.2.2). 

As a consequence, the discussion there applies to $\mathcal{D}(Y)$, hence also, locally, to $\mathcal{D}_X$. 

In particular, the $\mathcal{D}_X$-module is locally finitely generated if and only if each $x \in X$ has an affine neighborhood $Y$ such that $M(Y)$ has a good filtration $I$, with respect to $\mathcal{D}(Y)$. 



If so, we let again $I$ be the annihilator of $\mathrm{Gr} M(Y)$ in $\mathcal{O}(T^*(Y))$ and $J$ the radical of $I$. 

The variety in $T^*(Y)$ defined by $J$ is the characteristic variety of $M(Y)$. 

Since $J$ does not depend on the good filtration (V.2.2, lemma), this notion has an intrinsic character and allows one to define the {\color{red}characteristic variety}, or {\color{red}singular support} $\mathrm{SS}(M)$ of any coherent $\mathcal{D}_X$-module $M$. 

It is conical, in the sense that
\begin{equation}
\tau^{-1}(x) \cap \mathrm{SS}(M) \neq x \Longrightarrow \tau^{-1}(x) \subset \mathrm{SS}(M), \quad (x \in X).
\end{equation}

We let $d(M)$ be its dimension and, for $x \in X$, we let $d_x(M)$ be the dimension of $\mathrm{SS}(M) \cap \tau^{-1}(Y)$ for sufficiently small neighborhoods $Y$ of $x$. 

Let also
\[
j_x(M) = \min\{ j \mid \mathrm{Ext}^j_{\mathcal{D}_{X,x}}(M_x, \mathcal{D}_{X,x}) \ne 0 \}.
\]
It is equal to
\[
j(M(Y)) = \min\{ j \mid \mathrm{Ext}^j_{\mathcal{D}(Y)}(M(Y), \mathcal{D}(Y)) \ne 0 \}
\]
for sufficiently small affine neighborhoods $Y$ of $x$.

\end{frame}

